As someone who's stuck now - can't travel due to pets, I wish there were more pet sitters. There's one that does cover this area but I pretty sure if I told the person what pets I have, their very specific requirements and the fact that some will try to escape their cages, I would hear "uh... I don't do that".
If you really want to make it as a pet sitter, learn that pet doesn't only mean cats and dogs. Too many bird, reptile, fish, and even insect keepers are stuck without knowledgeable people to watch their pets. Unfortunately, and probably why most people won't do it, reptiles (lizards and turtles) need live insects and/or freshly prepared veggies.